10 INTEGRALS OF VECTOR-VALUED FUNCTIONS 155 and «// are integrable, and we have i ( from which it follows immediately that The set JS?c(X, A*) of /-i-integrable functions on X with values in C is a1 vector space over C, and/W fdfj, is a linear form on this vector space. Further- more, if/is any complex-valued integrable function, then |/| isintegrable and (13.10.3) sl I/I For since |/| = ((^/)2 + C//)2)1/2, it follows that |/| is measurable (13.9.6), and we have M*(l/l) ^ M*(l*/l) + /**(I>/D- Hence the first assertion, by (13.9.13). Also there exists a complex number C of absolute value 1 such that CK/) = IX/)I, and hence |/i(/)| = *WC/)) = M^(C/)) £ Kl/l), which establishes (13,10.3). If E' is the dual of the finite-dimensional vector space E, then to say that are integrable is equivalent to saying that, for each *' e E', the mapping ><f(.x),z'> is integrable (because it is a linear combination of the jQ. We generalize this as follows: if I is any set, then a mapping x\-*fx of X into the vector space K! (where K = R or C) is scalarly ^.-integrable if for each a e I the function x\->fx(a) is integrable. (13.10.4) LetEbeaFrdchetspace, E' its dual ([IAS). Let x^^ fx be a scalarly integrable mapping of X into E' such that, for each convergent sequence (art) in E, there exists a function g ^ 0 defined on X such that n*(g) < + oo and lfx(an)l ^ ff(x)for all n, almost everywhere in X. Then there exists a continuous linear form z' e E' on E such that <z, z'> = J fx(z) d^(x)for all t e E. By (3.13.4) it is enough to show that if (aw) is a sequence in E with limit z then f fJXz) dn(x)= lim f f,(an) ^u(jc); and this follows from (13.8.4). J M-'-OO*' The linear form z' so defined is called the integral (or weak integral) of the function x^>fx with respect to /^, and is denoted by | fx d^(x). Hence, forsets, the number